Quantization of measure in gravitation

TL;DR

This paper demonstrates that measures in gravitation can be quantized using the Radon-Nikodym theorem, leading to variable fundamental lengths and potential singularity smoothing, extending loop quantum gravity methods.

Contribution

It introduces a novel approach to quantizing measures in gravitation, not necessarily linked to the metric, using Radon-Nikodym derivatives and quantum measure concepts.

Findings

Variable fundamental length emerges from measure quantization.

Quantization of non-metric measures is possible in gravitation.

Potential smoothing of singularities through quantum measure.

Abstract

Using the Radon-Nikodym theorem concerning the relation between any two measures, as well as the methods employed in loop quantum gravity, it is shown that, in gravitation, one can quantize any measure which is not even associated with metric. We have considered the simplest case where the proportionality coefficient between two operators of measure (the Radon-Nikodym derivative) is some function. The result is that in the right-hand side of the commutation relations for the measure the fundamental length becomes a variable quantity, and this can lead to smoothing the singularity. The case where the Radon-Nikodym derivative is an operator is also under discussion. Classical and quantum theories on the background of space endowed with quantum measure are under consideration.